Dynamical simulation of wave solutions for the M-fractional Lonngren-wave equation using two distinct methods

This study integrates the (1 + 1)-dimensional time M-fractional Lonngren-wave (tM-fLW) equation. This equation is useful to describe the phenomenon of the electric signals in telegraph lines at the source of the tunnel diode. To extract some soliton solutions, two reliable analytical techniques, the new form of modified Kudryashov’s and the simplest equation, are used to accomplish this job. By implementing these techniques, the obtained solutions are expressed as rational, exponential, trigonometric, and hyperbolic functions with some free constraints. For the special values of the constraints, we obtained w-shape wave, bell-shaped wave, linked rogue wave, and multi-bell wave solutions by the NMK method, and dark bell and bright bell-shaped wave, linked rogue wave, periodic rogue wave, and w-shape wave by the SE method. The graphical depiction analyzes these traits, demonstrating the validity and efficacy of the suggested techniques. The obtained solutions, which have never been done before and show a good balance between the nonlinear physical components, are what make this research novel. It is remarkable to perceive that the simplest equation technique and the new form of modified Kudryashov’s technique are relaxed, companionable, and authoritative mathematical tools to elucidate a non-linear model. The movement role of the waves is explored and the modulation instability analysis is used to discuss the stability analysis of the obtained solutions, confirming that all created solutions are accurate and stable.